This paper gives an overview of the geometrization conjecture and approaches to its proof.

This paper characterizes those pseudo-Anosov mappings whose entropy can be detected homologically by taking a limit over finite covers. The proof is via complex-analytic methods. The same methods show the natural map ...

This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials ...

Let f(z) = e2 i z +z2, where θ is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: • the Hausdorff dimension of the Julia set J(f) is ...

This paper investigates several dynamically defined dimensions for rational maps \(f\) on the Riemann sphere, providing a systematic treatment modeled on the theory for Kleinian groups. We begin by defining the radial Julia ...

In this paper we study branched coverings of metrized, simplicial trees F : T → T which arise from polynomial maps f : C → C with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms ...